Ferromagnetic ordering of energy levels for symmetric spin chains

Consider a chain of $L$ urns, each containing $n$ balls, with the exchange dynamics described above: at each bond $\\{x,x+1\\}$, a random number of balls is selected according to $\rho_{x,x+1}$ and exchanged uniformly between the neighboring urns. The dynamics preserves the total number of red balls. For $1\leq k\leq nL-1$, let $\gamma_k$ denote the spectral gap restricted to configurations with exactly $k$ red balls and $nL-k$ white balls.

Ferromagnetic ordering conjecture. The spectral gap $\gamma_k$ is independent of $k$; equivalently,

$$\gamma_k=\gamma_1,$$

where $\gamma_1$ is the spectral gap of the associated random walk.

This is an analogue of Aldous's spectral-gap conjecture for the symmetric exclusion process and reflects the hidden $\operatorname{SU}(2)$ symmetry of the symmetric urn-mixing dynamics. The cases $k=0$ and $k=nL$ are excluded because the corresponding state spaces contain only one configuration and hence have no spectral gap.